Assessing evidence and testing appropriate hypotheses

It is crucial to identify the most appropriate hypotheses if one is to apply probabilistic reasoning to evaluate and properly understand the impact of evidence. Subtle changes to the choice of a prosecution hypothesis can result in drastically different posterior probabilities to a defence hypothesis from the same evidence. To illustrate the problem we consider a real case in which probabilistic arguments assumed that the prosecution hypothesis "both babies were murdered" was the appropriate alternative to the defence hypothesis "both babies died of Sudden Infant Death Syndrome (SIDS)". Since it would have been sufficient for the prosecution to establish just one murder, a more appropriate alternative hypothesis was "at least one baby was murdered". Based on the same assumptions used by one of the probability experts who examined the case, the prior odds in favour of the defence hypothesis over the double murder hypothesis are 30 to 1. However, the prior odds in favour of the defence hypothesis over the alternative 'at least one murder' hypothesis are only 5 to 2. Assuming that the medical and other evidence has a likelihood ratio of 5 in favour of the prosecution hypothesis results in very different conclusions about the posterior probability of the defence hypothesis.